The Gessel correspondence and the partial γ-positivity of the Eulerian polynomials on multiset Stirling permutations

作者：Chen, William Y. C.^*; Fu, Amy M.; Yan, Sherry H. F.

来源：European Journal of Combinatorics, 2023, 109: 103655.

DOI：10.1016/j.ejc.2022.103655

Pondering upon the grammatical labeling of 0-1-2 increasing plane trees, we come to the realization that the grammatical labels play a role as records of chopped off leaves of the orig-inal increasing binary trees. While such an understanding is purely psychological, it does give rise to an efficient apparatus to tackle the partial gamma-positivity of the Eulerian polynomials on multiset Stirling permutations, as long as we bear in mind the combinatorial meanings of the labels x and y in the Ges-sel representation of a k-Stirling permutation by means of an increasing (k+1)-ary tree. More precisely, we introduce a Foata- Strehl action on the Gessel trees resulting in an interpretation of the partial gamma-coefficients of the aforementioned Eulerian poly-nomials, different from the ones found by Lin-Ma-Zhang and Yan-Huang-Yang. In particular, our strategy can be adapted to deal with the partial gamma-coefficients of the second order Eulerian polynomials, which in turn can be readily converted to the combinatorial formulation due to Ma-Ma-Yeh in connection with certain statistics of Stirling permutations.

• 单位
  浙江师范大学; 天津大学